Chapter 2: Limits and Continuity

The number

*L*is the*limit of the function*(*f*) as*x**x*approaches*c*if, as the values of*x*get__arbitrarily close__(but not equal) to*c*, the values of*f*(*x*) approach (or equal)*L*.We write:

**lim**(*x*→*cf*)=*x**L*In order for

**lim**(*x*→*cf*) to exist, the values of*x**f*must tend to the same number*L*as*x*approaches*c*from__either the left or the right__.

We write **limx→c−f(x)** or the ** left-hand** limit of f at c (as x approaches c through values less than c).

We write **lim x→c**+

**Example:**

Prove that lim*x*→0|*x*|=0.

**SOLUTION:**

The function

*f*(*x*) is said to*become infinite*(positively or negatively) as*x*approaches*c*if*f*(*x*) can be made arbitrarily large (positively or negatively) by taking*x*sufficiently close to*c*.

We write **lim x→cf****(

Since, for the limit to exist, it must be a finite number, neither of the preceding limits exists.

This definition can be extended to

__include__*x*__approaching__*c*__from the left or from the right__. The following examples illustrate these definitions.

**Example:**

Describe the behavior of *f*(*x*)=1/*x* near *x* = 0 using limits.

SOLUTION:

We write:

**lim x→∞f**(

if the difference between *f*(*x*) and *L* can be made arbitrarily small by making *x* sufficiently large positively (or negatively).

**Example:**

SOLUTION:

The line

*y*=*b*is a*horizontal asymptote*of the graph of*y*=*f*(*x*) if**limx→∞f(x)=borlimx→−∞f(x)=b.**

Note, unlike vertical asymptotes, horizontal asymptotes can be crossed.

**Example:**

SOLUTION:

We see that *y* = 2 is a horizontal asymptote, since

lim*x*→+*∞k*(*x*)=lim*x*→−*∞k*(*x*)=2

Also, *x* = 3 is a vertical asymptote; the graph shows that

lim*x*→3−*k*(*x*)=−*∞* and lim*x*→3+*k*(*x*)=+*∞*

If *B*, *D*, *E*, *c*, and *k* are real numbers and if the limits of functions *f* and *g* exist at *x* = *c* such that lim*x*→*cf*(*x*)=*B* and lim*x*→*cg*(*x*)=*D* then:

**The Constant Rule**: lim*x*→*ck*=*k***The Constant Multiple Rule**: lim*x*→*ck*⋅*f*(*x*)=*k*⋅lim*x*→*cf*(*x*)=*k*⋅*B***The Sum Rule**: lim*x*→*c*(*f*(*x*)+*g*(*x*))=lim*x*→*cf*(*x*)+lim*x*→*cg*(*x*)=*B*+*D***The Difference Rule**: lim*x*→*c*(*f*(*x*)−*g*(*x*))=lim*x*→*cf*(*x*)−lim*x*→*cg*(*x*)=*B*−*D***The Product Rule**: lim*x*→*c*(*f*(*x*)⋅*g*(*x*))=lim*x*→*cf*(*x*)⋅lim*x*→*cg*(*x*)=*B*⋅*D***The Quotient Rule**:**The Composition Rule**:**The Squeeze or Sandwich Theorem**:

Squeezingfunctiongbetween functionsfandhforcesgto have the same limitLatx=cas dofandg.

**Example:**

To find

where *P*(*x*) and *Q*(*x*) are polynomials in *x*, we can divide both numerator and denominator by the highest power of *x* that occurs and use the fact that

Example:

**The Rational Function Theorem**

This theorem holds also when we replace “*x*→∞” by “*x*→−∞.”

Note also that:

2.

3.

**Example:**

**Example:**

Solution:

The value of *e* can be approximated on a graphing calculator to a large number of decimal places by evaluating

for large values of *x*.

If a function is continuous over an interval, we can draw its graph without lifting pencil from paper. The graph has no holes, breaks, or jumps on the interval.

Conceptually, if

*f*(*x*) is continuous at a point*x*=*c*, then the closer*x*is to*c*, the closer*f*(*x*) gets to*f*(*c*). This is made precise by the following definition:

**Definition**

The function *y* = *f*(*x*) is continuous at *x* = *c* if

*f*(*c*) exists (that is,*c*is in the domain of*f*)lim

*x*→*cf*(*x*) existslim

*x*→*cf*(*x*)=*f*(*c*)

A function is

**continuous**over the closed interval [*a*,*b*] if it is continuous at each*x*such that*a*⩽*x*⩽*b*.A function that is

**not continuous**at*x*=*c*is said to be discontinuous at that point. We then call*x*=*c*a*point of discontinuity*.

Polynomials are continuous everywhere—namely, at every real number.

The

**absolute-value function***f*(*x*) = |*x*|is continuous everywhere.The

**trigonometric, inverse trigonometric, exponential, and logarithmic functions**are continuous at each point in their domains.Functions of the type

*n*√*x*(where*n*is a positive integer ⩾ 2) are continuous at each*x*for which*n*√*x*is defined.The

**greatest-integer function**f(x) = [x] is discontinuous at each integer, since it does not have a limit at any integer.

The graph of *f* is shown at the right.

We observe that *f* is not continuous at *x* = −2, *x* = 0, or *x* = 2.

At *x* = −2, *f* is not defined.

At *x* = 0, *f* is defined; in fact, *f*(0) = 2.

However, since lim

*x*→0−*f*(*x*)=1 and lim*x*→0+*f*(*x*)=0, lim*x*→0*f*(*x*) does not exist.Where the left- and right-hand limits exist, but are different, the function has a

.*jump discontinuity*The greatest-integer function,

*y*= [*x*], has a jump discontinuity at every integer.

At *x* = 2, *f* is defined; in fact, *f*(2) = 0. Also, lim*x*→2*f*(*x*)=−2; the limit exists.

However, lim

*x*→2*f*(*x*)≠*f*(2).This discontinuity is called

.*removable*If we were to redefine the function at

*x*= 2 to be*f*(2) = −2, the new function would no longer have a discontinuity there.We cannot, however, “remove” a jump discontinuity by any redefinition whatsoever.

Whenever the graph of a function *f*(*x*) has the line *x* = *a* as a vertical asymptote, then *f*(*x*) becomes positively or negatively infinite as *x* → *a*+ or as *x* → *a*−.

The function is then said to have an

.*infinite discontinuity*

**Example:**

SOLUTION:

This function is continuous except where the denominator equals 0 (where *g* has an infinite discontinuity). It is not continuous at *x* = 3, but is continuous at *x* = 0.

The number

*L*is the*limit of the function*(*f*) as*x**x*approaches*c*if, as the values of*x*get__arbitrarily close__(but not equal) to*c*, the values of*f*(*x*) approach (or equal)*L*.We write:

**lim**(*x*→*cf*)=*x**L*In order for

**lim**(*x*→*cf*) to exist, the values of*x**f*must tend to the same number*L*as*x*approaches*c*from__either the left or the right__.

We write **limx→c−f(x)** or the ** left-hand** limit of f at c (as x approaches c through values less than c).

We write **lim x→c**+

**Example:**

Prove that lim*x*→0|*x*|=0.

**SOLUTION:**

The function

*f*(*x*) is said to*become infinite*(positively or negatively) as*x*approaches*c*if*f*(*x*) can be made arbitrarily large (positively or negatively) by taking*x*sufficiently close to*c*.

We write **lim x→cf****(

Since, for the limit to exist, it must be a finite number, neither of the preceding limits exists.

This definition can be extended to

__include__*x*__approaching__*c*__from the left or from the right__. The following examples illustrate these definitions.

**Example:**

Describe the behavior of *f*(*x*)=1/*x* near *x* = 0 using limits.

SOLUTION:

We write:

**lim x→∞f**(

if the difference between *f*(*x*) and *L* can be made arbitrarily small by making *x* sufficiently large positively (or negatively).

**Example:**

SOLUTION:

The line

*y*=*b*is a*horizontal asymptote*of the graph of*y*=*f*(*x*) if**limx→∞f(x)=borlimx→−∞f(x)=b.**

Note, unlike vertical asymptotes, horizontal asymptotes can be crossed.

**Example:**

SOLUTION:

We see that *y* = 2 is a horizontal asymptote, since

lim*x*→+*∞k*(*x*)=lim*x*→−*∞k*(*x*)=2

Also, *x* = 3 is a vertical asymptote; the graph shows that

lim*x*→3−*k*(*x*)=−*∞* and lim*x*→3+*k*(*x*)=+*∞*

If *B*, *D*, *E*, *c*, and *k* are real numbers and if the limits of functions *f* and *g* exist at *x* = *c* such that lim*x*→*cf*(*x*)=*B* and lim*x*→*cg*(*x*)=*D* then:

**The Constant Rule**: lim*x*→*ck*=*k***The Constant Multiple Rule**: lim*x*→*ck*⋅*f*(*x*)=*k*⋅lim*x*→*cf*(*x*)=*k*⋅*B***The Sum Rule**: lim*x*→*c*(*f*(*x*)+*g*(*x*))=lim*x*→*cf*(*x*)+lim*x*→*cg*(*x*)=*B*+*D***The Difference Rule**: lim*x*→*c*(*f*(*x*)−*g*(*x*))=lim*x*→*cf*(*x*)−lim*x*→*cg*(*x*)=*B*−*D***The Product Rule**: lim*x*→*c*(*f*(*x*)⋅*g*(*x*))=lim*x*→*cf*(*x*)⋅lim*x*→*cg*(*x*)=*B*⋅*D***The Quotient Rule**:**The Composition Rule**:**The Squeeze or Sandwich Theorem**:

Squeezingfunctiongbetween functionsfandhforcesgto have the same limitLatx=cas dofandg.

**Example:**

To find

where *P*(*x*) and *Q*(*x*) are polynomials in *x*, we can divide both numerator and denominator by the highest power of *x* that occurs and use the fact that

Example:

**The Rational Function Theorem**

This theorem holds also when we replace “*x*→∞” by “*x*→−∞.”

Note also that:

2.

3.

**Example:**

**Example:**

Solution:

The value of *e* can be approximated on a graphing calculator to a large number of decimal places by evaluating

for large values of *x*.

If a function is continuous over an interval, we can draw its graph without lifting pencil from paper. The graph has no holes, breaks, or jumps on the interval.

Conceptually, if

*f*(*x*) is continuous at a point*x*=*c*, then the closer*x*is to*c*, the closer*f*(*x*) gets to*f*(*c*). This is made precise by the following definition:

**Definition**

The function *y* = *f*(*x*) is continuous at *x* = *c* if

*f*(*c*) exists (that is,*c*is in the domain of*f*)lim

*x*→*cf*(*x*) existslim

*x*→*cf*(*x*)=*f*(*c*)

A function is

**continuous**over the closed interval [*a*,*b*] if it is continuous at each*x*such that*a*⩽*x*⩽*b*.A function that is

**not continuous**at*x*=*c*is said to be discontinuous at that point. We then call*x*=*c*a*point of discontinuity*.

Polynomials are continuous everywhere—namely, at every real number.

The

**absolute-value function***f*(*x*) = |*x*|is continuous everywhere.The

**trigonometric, inverse trigonometric, exponential, and logarithmic functions**are continuous at each point in their domains.Functions of the type

*n*√*x*(where*n*is a positive integer ⩾ 2) are continuous at each*x*for which*n*√*x*is defined.The

**greatest-integer function**f(x) = [x] is discontinuous at each integer, since it does not have a limit at any integer.

The graph of *f* is shown at the right.

We observe that *f* is not continuous at *x* = −2, *x* = 0, or *x* = 2.

At *x* = −2, *f* is not defined.

At *x* = 0, *f* is defined; in fact, *f*(0) = 2.

However, since lim

*x*→0−*f*(*x*)=1 and lim*x*→0+*f*(*x*)=0, lim*x*→0*f*(*x*) does not exist.Where the left- and right-hand limits exist, but are different, the function has a

.*jump discontinuity*The greatest-integer function,

*y*= [*x*], has a jump discontinuity at every integer.

At *x* = 2, *f* is defined; in fact, *f*(2) = 0. Also, lim*x*→2*f*(*x*)=−2; the limit exists.

However, lim

*x*→2*f*(*x*)≠*f*(2).This discontinuity is called

.*removable*If we were to redefine the function at

*x*= 2 to be*f*(2) = −2, the new function would no longer have a discontinuity there.We cannot, however, “remove” a jump discontinuity by any redefinition whatsoever.

Whenever the graph of a function *f*(*x*) has the line *x* = *a* as a vertical asymptote, then *f*(*x*) becomes positively or negatively infinite as *x* → *a*+ or as *x* → *a*−.

The function is then said to have an

.*infinite discontinuity*

**Example:**

SOLUTION:

This function is continuous except where the denominator equals 0 (where *g* has an infinite discontinuity). It is not continuous at *x* = 3, but is continuous at *x* = 0.